TITLE: Notations for dual spaces and dual operators
QUESTION [9 upvotes]: I'm asking for opinions about the 'best' notations for:
 1. the algebraic dual of a vector space $X$;
 2. the continuous dual of a TVS;
 3. the algebraic dual (transpose) of an operator $T$ between vector spaces;
 4. the dual (transpose) of a continuous operator between TVS;
 5. the adjoint of a bounded operator $T$ between Hilbert spaces.
My problem is that I would like to use these notions in the same context. The standard notations tend to overlap but I am forced to use different notations for each of these entities. Of course it is easy to come up with notations, but some traditions are well established and it is not trivial to respect them and at the same time keep them apart, with some elegance.
What I'm using now:
 1. $X'_{alg}$
 2. $X'$
 3. $^tT$
 4. $T'$
 5. $T^*$
Thank you for your advice.

REPLY [4 votes]: Given your situation of having to juggle all these notational traditions at the same time, I would recommend for a space $X$ and an operator $A$:

$X^{\vee}$, 2. $X'$, 3. $A^{\rm T}$, 4. $A^{\rm T}$, 5. $A^*$

My rationale is as follows. 
For 1: Algebraic geometers especially use the "vee' notation for the algebraic dual so I think the cultural association helps the brain automatically make the association with the algebraic notion of dual.
For 2: One usually does not write the space of tempered distributions as $S^*(\mathbb{R}^d)$ but rather as $S'(\mathbb{R^d})$. 
For 3 and 4: I would use the same notation as per the comment by user95282 since the maps are related by restriction. One could use ${}^{\rm t}A$ for both but I prefer the matrix algebra notation if only because it is easier to type.
For 5: As yuggib said, it is standard in the theory of $C^*$-algebras and spectral theory. I don't see any reason to be a contrarian and not follow what everyone else does.